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重难点16数列的综合应用十二大题型(原卷版)_11.4-2_2026-11-04

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重难点16 数列的综合应用【十二大题型】

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M(N)ŊŋM(N + O) = M(N)M(O)
	M(N)
	M(O) +6N
M(1) = 4NÝM(1) +M(2) +  +M(99) = ń   ň
Aƒ299 +198
Bƒ299 +196
Cƒ2100 +198
Dƒ2100 +196
ą, 7-1ń2024·ȼȽ·ï¬ňÎ
"{}#r n qsçNK  N)Ï6
MN = ,N2
+GN + P,GP  Q#ĂăÒNÝń  ň
AƒP0 = 1
BƒÜ, = 0NÝR0  NNÆêT
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DƒÜ, < 0NÝR0  NNÆêT
ą, 7-2ń2024·ÒŒ·¬ŎŏŐňĀeM(N) =
1
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2NN
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6
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"{}#pq,ª
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4N
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2025 (  N*)N€
"{}#r 2024 qs2024.
ą, 7-3ń2024·ǻǼ·[¬ňĀe
"{}#r n qsçNn çċČ
Nő3(
) = 4(
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 22
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W > 1NN > 
őN C 0N6
MN = (1 + NW
WN
ƒ
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MN > 0ª
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ln(1 + N < Nª
(3)Î  #N23
1 + 2
1
2 + 3
1
3 + + 
1
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ln + 1)ƒ
ą, 8-1ń2024·ǻǼǼȾ·ø¬ňĀe”
4  QNÎM(N) = lnN +
4
NN
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O = M(N)Ï(1,1)ij#ȿɀŒª
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2
>
M(N1)
M(N2)
N1
N2
ƒ
ą, 8-2ń2024·ljŚ·[¬ňĀe > 0Nő C 1N6
M(N) = N + ln(1 + N)
.
(1)Ɔ = M()
ln( + 1) +ç
"{}#rqs.23
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8
9N64 < 2024ª
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1
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#Řëùk.
      
      
ą, 8-3ń2024·śƪɁɂ·¬ŎŏŐňĀe6
M(N) = ln(N + 1)T(N) =
N
N2Nľ' 6 1ƒ
(1)Ü = 1N23
N > 0NM(N) <
N
	T(N21)ª
(2)Ü6
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#ëª
(3)Āe
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 N€2
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97
132
Bƒ
49
132
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47
66
Dƒ
49
66
ą, 9-1ń2024·ljǼɌɍ·[¬ňÜ
"{}#pq,ç = 

NƆÏ
"{}#r + 2(  N)
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3
BƒI9 < I10
CƒI10 < I11
DƒI11 < I12
ą, 9-2ń2024·ǻǼǻǚ·¬ŎŏŐňɉ Ɏ ɏøɐ¡¢ƝɑɒɓNdžŮɔɕ[ɖoġVɗ#ċŒĕ
ɘ4²CƝɑ#Œ,
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"{W}#pq,ª
      
      
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4
3.
ą, 9-3ń2024·Ōō·¬ŎŏŐňɉ Ɏ–ƃUɛɜNdžɐə'Ɗ0 3 ɝɃ_ƛËɞɟNľ'ɉə'# 3
ɝɞɟç 1 ɝǟɠs 2 ɝɡɠNɎə'# 3 ɝɞɟǔIǟɠ#NxϖɐƊ]Ċĭ#ɞɟ'ɢǀŘ 1 ɝN
t5‘Œ$ɣNźLjٜS#ɤĉNƆɉə'ɡɠƛËɞɟNɝNɥ0 2 ɝɡɠƛËɞɟ#89çWNɥ
0 1 ɝɡɠƛËɞɟ#89çBƒ
(1)€W2,B2#ëƒ
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(3)Ɔ = 2W + Bƒ
ńiň23
"{
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 10  
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4N5 = 44
ƒ
ń1ň€ƒ
ń2ňÏ<=ƮƯɨǕŠN+O'NÎK_A(AA)A = 1,2,3  )NƮɀ_A_A+1#ɩ9ç2ANő1 = 1N€
"{}
#pq,ƒ
ą, 10-1ń2024·úŅÛǚ·ï¬ňĀeɪɀ`:O2 = 	WNW > 0)NƮɀaDO = AN
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bP+^,b^+,.
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ą, 10-2ń2024·śƪHɫ·ï¬ňĀe{}IƊqVçċ
#!
"Nő1 + 2 = 3N3
2 = 2N

"{}#rqsçNő3 = 5N4 = 16.
ń1ň€
"{}N{}#pq,ª
ń2ňßĂÏ<=ƮƯɨǕŠ'NKI1(1,0)NI2(2,0)N…I(,0)NI+1(+1,0)N
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"{}#rqs.
ą, 10-3ń2024·úŅLɄ·¬ŎŏŐňĀeɬȄPDN2
2 + O2
2 =  >  > 0)NKIIɬȄÒ#ĦKN\1,\2I
ɭ ɮɯKNLI b I\1\2#ź¶NőL´K^ 
 1
3 ,0 5K# 1
3 ,0 #ɰæZsç
4
3ƒ
(1)€ɬȄ C #Œª
(2)ÜƮɀ l ÈKI(4,0)N5ɬȄ$X ANB –KƒÜ|,I|,|,G|,|GI|.!
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(1)
"{}#pq,ª
(2)#êUëNƌ€įŘÇêUë n #ë.
ą, 11-1ń2024·ȻŒŚȽ·ï¬ňĀe
"{}N_______________.ș]Ə"–ÊÄî'ŗ[ÊNɳ
ɴÏÒ=#&'ƌ1h.ń
ßƱÞQÊÄîN¼¾í[Ê1hÉ·.ň%
"{}#rqsç = 2

	ń  #ňª&
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ń  #ň.
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(2)ȩ =  + log2N€
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ą, 11-2ń2024·úŅƖŇ·ø¬ňĀe{}I
"N{}I!
"Nő{}#rqsçN21 =
1 = 2N5 = 5(4
3)NÏ%5 = 4(4
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#1h.
(1)€
"{}s{}#pq,ª
(2)Î
" 
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ą, 11-3ń23-24 ïƏ·ťȲɵɶ·ķſňĀe
"{}#rqsçNő4 = 10,3 = 18.
(1)€
"{}#pq,ª
(2)ÜƊqVçċ
#
"{}ľrqsçNĩ]Äî% Äî& ÄîdœøÊÄî'Þ[Êĉç
ĀeNÎ =  + N€
"{}#pq,s
"{}#rqs^.
Äî%
 = 3
ª
Äî&
1 = 2,
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J 6 2ő  Zǔ02
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 12  
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Ń 12ń2024·ƶťɝŲɷ·ï¬ňßƱq
˜Ć#
"{},{}Ŋŋ{} f {} = {	
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N: < :ª:çɹ
N: > :Nľ':  {	
}NǖǗɦş{},{}ç“™ɳ$ɺ
"”NƆ 1,2
1,2
ç{},{}#“™ɳ$ɺ
"‘”Nç{}#rqs.
(1)Ü{} f {} = {1,2,3,4,5,6}Nő1 = 5NȚwØ0ŊŋÄî#“™ɳ$ɺ
"‘"ª
(2)į{},{}ç“™ɳ$ɺ
"”N
ńiň23
ŘêTëNǪÏ: = 	ª
ńiiňįçɹ
N€#êTë.
ą, 12-1ń2024·ɻɄ·¬ŎŏŐňĀeċČ
4NÎ1N2N…N	4N1N2N…N	4I4Êɼɽ
(
N =
	4
:=1: =
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A.(4)—ɾĢ
ò.
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NÆÇœ 8 Ê
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òª
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(3)Ü4 6 4Nő1N2N…N	4N1N2N…N	4A.( 4)—ɾĢ
òN€:(: = 1,2,  	4)#êT
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ä,
ńiň(N1 + N2 + N3)2 6 3(N1N2 + N2N3 + N3N1)NįőɿįN1 = N2 = N3Řªńiiňįċɹ
 6 4
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"{},{}NßƱǪÏċČ
0 6 3NẐċČ
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